The design of non slender column according to Eurocode 2 is discussed in this article. This article guides the design procedures to be followed.
Brace Non-Slender Column Design
- Edge column
- 300mm square column
- Axial Load 1500kN
- Moment at top -40kNm
- Moment at Bottom 45kNm
- fck 30N/mm2
- fyk 500N/mm2
- Nominal Cover 25mm
- Floor to Floor height 4250mm
- Depth of the beam supported by the column 450mm
Mtop = -40kNm
Mbottom = 45kNm
NEd = 1500kN
Clear height = 4250-450
= 3800mm
Effective length = lo
= factor * l
Factor = 0.85 (concise Eurocode 2, Table 5.1. This may more conservative).
lo = 0.85* 3800
= 3230mm
Slenderness λ = lo/i
i = radios of gyration = h/√12
λ = lo/( h/√12 ) = 3.46*lo/h = 3.46*3230/300 = 37.3
Limiting Slenderness λlim
λlim = 20ABC/√n
A = 0.7 if effective creep factor is unknown
B = 1.1 if mechanical reinforcement ratio is unknown
C = 1.7 – rm = 1.7-Mo1/Mo2
Mo1 = -40kNm
Mo2 = 45kNm where lMo2l ≥ lMo1l
C = 1.7 – (-40/45) = 2.9
n = NEd / (Ac*fcd)
fcd = fck / 1.5 = (30/1.5)*0.85 = 17
n = 1500*1000 / (300*300*17)= 0.98
λlim = 20*0.7*1.1*2.9/√0.98 = 45.1
λlim > λ hence, column is not slender.
Calculation of design moments
MEd = Max{Mo2, MoEd +M2, Mo1 + 0.5M2}
Mo2 = Max {Mtop, Mbottom} + ei*NEd = 45 + (3.23/400)*1500 ≥ Max(300/30, 20)*1500 = 57.1kNm > 30kNm
Mo2 = Min{Mtop, Mbottom} + ei*NEd = -40 + (3.23/400)*1500 ≥ Max(300/30, 20)*1500 = 27.9kNm
MoEd = 0.6*Mo2+ 0.4*Mo1 ≥ 0.4*Mo2 = 0.6*57.1 + 0.4*(-27.9) ≥ 0.4*57.1 = 23.1 ≥ 22.84
M2 = 0 , Column is not slender
MEd = Max{Mo2, MoEd +M2, Mo1 + 0.5M2}= Max{57.1, 23.1 +0, -27.9 + 0.5*0} = 57.1kNm
MEd / [b*(h^2)*fck] = (57.1*10^6) / [300*(300^2)*30] = 0.07
NEd / (b*h*fck) = (1500*10^6) / (300*300*30 = 0.56
Assume 25mm diameter bars as main reinforcement and 10mm bars as shear links
d2 = 25+10+25/2 = 47.5mm
d2/h = 47.5 / 300 = 0.16
Note: d2/h = 0.20 chart is reffed to find the reinforcement area, but it is more conservative. Interpolation can be used to find the exact value.
As*fyk / b*h*fck = 0.24
As = 0.24*300*300*30 / 500 = 1296mm2
Provides four 25mm bars (As Provided 1964mm2)
Check for Biaxial Bending
Further check is not required if
0.5 ≤ ( λy/ λz) ≤ 2.0 For rectangular column
and
0.2 ≥ (ey/heq)/(ez/beq) ≥ 5.0
Here λy and λz are slenderness ratios
λy is nearly equal to λz
Therefore, λy/λz is nearly equal to one.
Hence, λy/λz < 2 and > 0.5 OK
ey/heq = (MEdz / NEd) / heq
ez/beq = (MEdy / NEd) / beq
(ey/heq)/(ez/beq) = MEdz / MEdy Here h=b=heq=beq, column is square
MEdz = 45kNm
MEdy = 30kNm
Minimum moment, see the calculation of Mo2 for the method of calculation note: Moments due to imperfections need to be included only in the direction where they have the most unfavorable effect – Concise Eurocode 2
(ey/heq)/(ez/beq) = 45/30
= 1.5 > 0.2 and < 5
Therefore Biaxial check is required.
(MEdz / MRdz)^a + (MEdy / MRdy)^a ≤ 1
MEdz = 45kNm
MEdy = 30kNm
MRdz and MRdy are the moment resistance in the respective directions, corresponding to an axial load NEd.
For symetric reinforcement section
MRdz = MRdy
As Provided = 1964mm2
As*fyk / b*h*fck = 1964*500/(300*300*30) = 0.36
NEd / (b*h*fck) = 0.56
From the chart d2/h =0.2
MEd / [b*(h^2)*fck] = 0.098
MEd = 0.098*300*300*300*30 = 79.38kNm
a = an exponent
a = 1.0 for NEd/NRd = 0.1
a = 1.5 for NEd/NRd = 0.7
NEd = 1500kN
NRd = Ac*fcd + As*fyd
NRd = 300*300*(0.85*30/1.5) + 1964*(500/1.15) = 2383.9kN
NEd/NRd = 1500/2383.9
= 0.63
By interpolating
a = 1.44
(MEdz / MRdz)^a + (MEdy / MRdy)^a = (45 / 79.39)^1.44 + (30 / 79.38)^1.44 = 0.69 <1
Hence, Check for biaxial bending is ok
Therefore, Provide four 25mm diameter bars.
